# The Weighted Surplus Division Value for Cooperative Games

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- Efficiency: For any game $v\in {\mathcal{G}}^{N}$, ${\sum}_{i\in N}{\varphi}_{i}\left(v\right)=v\left(N\right)$.
- Symmetry: For any game $v\in {\mathcal{G}}^{N}$, if players $i,j\in N$ are symmetric, then ${\varphi}_{i}\left(v\right)={\varphi}_{j}\left(v\right)$.
- Individual rationality: For any weakly essential game $v\in {\mathcal{G}}^{N}$, ${\varphi}_{i}\left(v\right)\ge v\left(\left\{i\right\}\right)$, for all $i\in N$.
- Additivity: For any games $v,w\in {\mathcal{G}}^{N}$, $\varphi (v+w)=\varphi \left(v\right)+\varphi \left(w\right)$.
- Linearity: For any games $v,w\in {\mathcal{G}}^{N}$ and $p,q\in \mathbb{R}$, $\varphi (pv+qw)=p\varphi \left(v\right)+q\varphi \left(w\right)$.
- Inessential game property: For any inessential game $v\in {\mathcal{G}}^{N}$, ${\varphi}_{i}\left(v\right)=v\left(\left\{i\right\}\right)$ for all $i\in N$.

## 3. The Weighted Surplus Division Value and Its Procedural Interpretation

**Definition**

**1.**

- Step 1
- Each player goes into a permutation $\pi $ randomly and all orders in $\Pi \left(N\right)$ have the same probability.
- Step 2
- Every arriving player, $i\in N$, joins the coalition of his predecessors to form a new coalition ${P}_{i}^{\pi}$ and brings his marginal contribution $v\left({P}_{i}^{\pi}\right)-v({P}_{i}^{\pi}\setminus \left\{i\right\})$ to the coalition of his predecessors.
- Step 3
- The arriving player $i\in N$ claims his individual worth $v\left(\right\{i\left\}\right)$ and the surplus (or deficit), $v\left({P}_{i}^{\pi}\right)-v({P}_{i}^{\pi}\setminus \left\{i\right\})-v\left(\left\{i\right\}\right)$, is shared (or afforded) by all players according to the weight system $\omega $.
- Step 4
- A player’s value is the expected payoff of his part of $v\left(N\right)$ (over all orders of arrival).

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Optimization Implementation

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

- Equal disweighted complaint property: For any game $v\in {\mathcal{G}}^{N}$ and weight vector $\omega \in {\u25b5}^{n}$, the value $\varphi :{\mathcal{G}}^{N}\to {\mathbb{R}}^{N}$ verifies that$${e}^{\omega}(i,\varphi )={e}^{\omega}(j,\varphi ),\mathrm{for}i,j\in N.$$

**Theorem**

**3.**

## 5. Axiomatizations

- ω-symmetry for zero-normalized game: For any zero-normalized game $v\in {\mathcal{G}}^{N}$ and weight vector $\omega \in {\u25b5}^{n}$, the value $\varphi :{\mathcal{G}}^{N}\to {\mathbb{R}}^{N}$ verifies that$$\frac{{\varphi}_{i}\left(v\right)}{{\omega}_{i}}=\frac{{\varphi}_{j}\left(v\right)}{{\omega}_{j}},\mathrm{for}\mathrm{any}i,j\in N.$$

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

**Corollary**

**3.**

**Proof.**

- (i)
- For any unanimity game ${u}_{T}$, $T\subseteq N$ and $t=1$, it follows that ${\sum}_{j\in N}{u}_{T}\left(\left\{j\right\}\right)\le {u}_{T}\left(N\right)$. According to the individual rationality,$$\left\{\begin{array}{cc}i\notin T,\hfill & \phantom{\rule{4pt}{0ex}}{\varphi}_{i}\left({u}_{T}\right)\ge {u}_{T}\left(\left\{i\right\}\right)=0,\hfill \\ i\in T,\hfill & \phantom{\rule{4pt}{0ex}}{\varphi}_{i}\left({u}_{T}\right)\ge {u}_{T}\left(\left\{i\right\}\right)=1.\hfill \end{array}\right.$$With efficiency, we obtain that$${\varphi}_{i}(N,{u}_{T})=\left\{\begin{array}{cc}1,\hfill & \phantom{\rule{4pt}{0ex}}i\in T,\hfill \\ 0,\hfill & \phantom{\rule{4pt}{0ex}}i\notin T.\hfill \end{array}\right.$$
- (ii)
- Given a standard game ${b}_{T}$, $T\subseteq N$ and $1<t<n$, it holds that ${\sum}_{j\in N}{b}_{T}\left(\left\{j\right\}\right)\le {b}_{T}\left(N\right)$. According to the individual rationality, for any $i\in N$, ${\varphi}_{i}\left({b}_{T}\right)\ge {b}_{T}\left(\left\{i\right\}\right)=0$. With efficiency, ${\varphi}_{i}\left({b}_{T}\right)=0$, $i\in \phantom{\rule{3.33333pt}{0ex}}N$.
- (iii)
- For $T=N$, the standard game ${b}_{N}$ is a zero-normalized game. According to the $\omega $-symmetry for zero-normalized game, we have, for any $i,j\in N$, $\frac{{\varphi}_{i}\left({b}_{N}\right)}{{\omega}_{i}}=\frac{{\varphi}_{j}\left({b}_{N}\right)}{{\omega}_{j}}$. With efficiency, ${\varphi}_{i}\left({b}_{N}\right)={\omega}_{i}$, $i\in \phantom{\rule{3.33333pt}{0ex}}N$.

**Remark**

**4.**

- Individual equity: For any game $v\in {\mathcal{G}}^{N}$, if ${\sum}_{j\in N}v\left(\left\{j\right\}\right)=v\left(N\right)$, then ${\varphi}_{i}\left(v\right)=v\left(\left\{i\right\}\right)$, for all $i\in \phantom{\rule{3.33333pt}{0ex}}N$.

**Corollary**

**4.**

- Covariance: For any game $v\in {\mathcal{G}}^{N}$, $p\in \mathbb{R}$ and $q\in {\mathbb{R}}^{N}$, ${\varphi}_{i}(pv+q)=p{\varphi}_{i}\left(v\right)+{q}_{i}$, for $i\in N$, where $(pv+q)\left(S\right)=pv\left(S\right)+{\sum}_{j\in S}{q}_{j}$, $S\subseteq N$.

**Corollary**

**5.**

**Proof.**

- (i)
- For any unanimity game ${u}_{\left\{k\right\}}$, $k\in N$, let$${u}^{0}\left(S\right):={u}_{\left\{k\right\}}\left(S\right)-\sum _{j\in S}{u}_{\left\{k\right\}}\left(\left\{j\right\}\right),S\subseteq N.$$For $i\in N$, ${u}^{0}\left(\left\{i\right\}\right)=0$, so ${u}^{0}$ is a zero-normalized game. Because of the $\omega $-symmetry for zero-normalized game,$$\frac{{\varphi}_{i}\left({u}^{0}\right)}{{\omega}_{i}}=\frac{{\varphi}_{j}\left({u}^{0}\right)}{{\omega}_{j}},\mathrm{for}\mathrm{any}i,j\in N.$$Combined with the efficiency, we can obtain$${\varphi}_{i}\left({u}^{0}\right)={\omega}_{i}{u}^{0}\left(N\right)=0,\mathrm{for}i\in N.$$Because ${u}_{\left\{k\right\}}\left(S\right)={u}^{0}\left(S\right)+{\sum}_{j\in S}{u}_{\left\{k\right\}}\left(\left\{j\right\}\right)$, $S\subseteq N$, with covariance, it holds that$${\varphi}_{i}\left({u}_{\left\{k\right\}}\right)={\varphi}_{i}\left({u}^{0}\right)+{u}_{\left\{k\right\}}\left(\left\{i\right\}\right)={u}_{\left\{k\right\}}\left(\left\{i\right\}\right),fori\in N.$$
- (ii)
- Given any standard game ${b}_{T}$, $T\subseteq N$ and $t\ge 2$, they are all zero-normalized games. According to the $\omega $-symmetry for zero-normalized game, we have, for any $i,j\in N$, $\frac{{\varphi}_{i}\left({b}_{T}\right)}{{\omega}_{i}}=\frac{{\varphi}_{j}\left({b}_{T}\right)}{{\omega}_{j}}$. With efficiency, ${\varphi}_{i}\left({b}_{T}\right)={\omega}_{i}{b}_{T}\left(N\right)$, $i\in N$.

**Example**

**1.**

- (i)
- If we divide the surplus division value $v\left(N\right)-{\sum}_{i\in N}v\left(\left\{i\right\}\right)$ equally, then the payoff coincides with the equal surplus division value $ES\left(v\right)$.
- (ii)
- If we allocate the surplus part according to a weight ${w}_{i},(i=1,2,3)$, the payoff coincides with the weighted surplus division value $S{D}^{\omega}\left(v\right)$. Naturally, the weight can be set to be the proportion of their investment.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Yang, H.; Wang, W.; Ding, Z.
The Weighted Surplus Division Value for Cooperative Games. *Symmetry* **2019**, *11*, 1169.
https://doi.org/10.3390/sym11091169

**AMA Style**

Yang H, Wang W, Ding Z.
The Weighted Surplus Division Value for Cooperative Games. *Symmetry*. 2019; 11(9):1169.
https://doi.org/10.3390/sym11091169

**Chicago/Turabian Style**

Yang, Hui, Wenna Wang, and Zhengsheng Ding.
2019. "The Weighted Surplus Division Value for Cooperative Games" *Symmetry* 11, no. 9: 1169.
https://doi.org/10.3390/sym11091169